Anisotropic Media in 2d and Their Thermo-elastic Properties

contains the elasticity modules of the medium. Usually one makes the assumption that S is positive definite such that A(D) is a positive and self-adjoint operator. Then the first equation is hyperbolic, while the second one is parabolic. In [2] the thermo-elastic system is considered and dispersive estimates are proven under the assumptions that (A1-2) A(ξ) is positive (self-adjoint, 2-homogeneous and real-analytic in η = ξ/|ξ|); (A3) #spec A(η) = 2 for all η ∈ S; (A4) γ 6= κj(η)− tr A(η) for all hyperbolic directions (with respect to κj(η) ∈ spec A(η)), where a direction η ∈ S is called hyperbolic with respect to κj(η) if the corresponding (normalised) eigenvector rj(η) is perpendicular to the direction η. It turns out that the decay properties of solutions depend only on the neighbourhood of hyperbolic directions. Essential are the